The matching, birthday and the strong birthday problem: a contemporary review

This article provides a contemporary exposition at a moderately quantitative level of the distribution theory associated with the matching and the birthday problems. A large number of examples, many not well known, are provided to help a reader have a feeling for these questions at an intuitive level.     

New simulation results for the calibration and inverse median estimation problems

This paper gives the results of a new simulation study for the familiar calibration problem and the less familiar inverse median estimation problem. The latter arises when one wishes to estimate from a linear regression analysis the value of the independent variable corresponding to a specified value of the median of the dependent variable. For example, from the results of a regression analysis between stress and time to failure, one might wish to estimate the stress at which the median time to failure is 10,000 hours. In the study, the mean square error, Pitman closeness, and probability of overestimation are compared for both the calibration problem and the inverse median estimation problem for (1) the classical estimator, (2) the inverse estimator, and (3) a modified version of an estimator proposed by Naszodi (1978) for both a small sample and a moderately large sample situation.     

Justifying an Invalid Derivation of the Probability of a Union of Events

An appealing, but invalid, derivation of the probability that at least one of n events occurs is justified, using a particular definition of subtraction of events. The probabilities that exactly m and at least m of the n events occur are derived similarly.     

Methods for Studying Generalized Birthday and Coupon Collection Problems

In this paper, we consider generalizations of two classical probability problems: the birthday problem and the coupon collector's problem. These problems are discussed in terms of urn models and captured through generating functions. Some methods for the study of the problems are presented. Furthermore, we also formulate the generalized birthday and coupon collector's problems as the waiting time problems. In each case, numerical examples are given in order to illustrate the feasibility of our methods.     

A general method of finding a minimax estimator of a distribution function when no equalizer rule is available

We propose a feasible general method for finding a minimax estimator of an unknown distribution function F in the nonparametric problem. As an application, some minimax estimators are proposed. Furthermore, some minimax binomial parametric problems are studied.     

Duality and local sensitivity analysis in least squares,minimax, and least absolute values regressions

Abstract

This paper deals with the problem of local sensitivity analysis in regression, i.e., how sensitive the results of a regression model (objective function, parameters, and dual variables) are to changes in the data. We use a general formula for local sensitivities in optimization problems to calculate the sensitivities in three standard regression problems (least squares, minimax, and least absolute values). Closed formulas for all sensitivities are derived. Sensitivity contours are presented to help in assessing the sensitivity of each observation in the sample. The dual problems of the minimax and least absolute values are obtained and interpreted. The proposed sensitivity measures are shown to deal more effectively with the masking problem than the existing methods. The methods are illustrated by their application to some examples and graphical illustrations are given.     

Testing for a change in covariance operator

The paper considers a problem of equality of two covariance operators. Using functional principal component analysis, a method for testing equality of K largest eigenvalues and the corresponding eigenfunctions, together with its generalization to a corresponding change point problem is suggested. Asymptotic distributions of the test statistics are presented.     

Influence functions related to eigenvalue problems which appear in multivariate analysis

Tanaka(1988) derived two influence functions related to an ordinary eigenvalue problem (A–λ_s I)v_s = 0 of a real symmetric matrix A and used them for sensitivity analysis in principal component analysis. One of these influence functions was used to develop sensitivity analysis in factor analysis (see e.g. Tanaka and Odaka, 1988a). The present paper derives some additional influence functions related to an ordinary eigenvalue problem and also several influence functions related to a generalized eigenvalue problem (A–θ_s A)u_s = 0, where A and B are real symmetric and real symmetric positive definite matrices, respectively. These influence functions are applicable not only to the case where the eigenvalues of interest are all simple but also to the case where there are some multiple eigenvalues among those of interest.     

A Note on the Uniformity Assumption in the Birthday Problem

If 23 people are selected at random from some large population whose birthdays are uniformly distributed throughout the year, then it is well-known that the chances are better than even that at least two of them will have the same birthday. In this note it is shown that this is true for any distribution of birthdays.     

A modification of the kruskal-wallis statistic for the generalized behrens-fisher problem

The problem of testing the equality of the medians of several populations is considered. Standard distribution-free procedures for this problem require that the populations have the same shape in order to maintain their nominal significance level, ever asymptotically, under the null hypothesis of equal medians , A modification of the Kruskal-Wallis test statistic is proposed which is exactly distribution-free under the usual nonparanetric asswnption that the continuous populations are identical with any shape. It is asymptotically distribution-free when the Continuous populations are asswned to be syrmnetric with equal medians.     

A stochastic procedure to solve linear ill-posed problems

In this work, we propose a stochastic procedure of Robbins–Monro type to resolve linear inverse problems in Hilbert space. We study the probability of large deviation between the exact solution and the approximated one and build a confidence domain for the approximated solution while precising the rate of convergence. To check the validity of our work, we give a simulation application into a deconvolution problem.     

The probability generating function of empty cell variable in a randomized occupancy problem

Let M_o denote the number of empty cells when n distinguishable balls are distributed independently and at random in ra cells such that each ball stays with probability p in its cell, and falls through with probability 1-p. We find the probability generating function of M_o by solving a partial differential equation satisfied by a suitable generating function. The corresponding function for the classical case p = 1 is well-known, but obtained by different methods.     

Some Nonparametric Asymptotic Results for a Class of Stochastic Processes

This article provides a solution of a generalized eigenvalue problem for integrated processes of order 2 in a nonparametric framework. Our analysis focuses on a pair of random matrices related to such integrated process. The matrices are constructed considering some weight functions. Under asymptotic conditions on such weights, convergence results in distribution are obtained and the generalized eigenvalue problem is solved. Differential equations and stochastic calculus theory are used.     

An inclusion-consistent solution to the problem of absurd confidence statements: 2. Consistent nonexact-confidence-interval estimation

Two consistent nonexact-confidence-interval estimation methods, both derived from the consistency-equivalence theorem in Plante (1991), are suggested for estimation of problematic parametric functions with no consistent exact solution and for which standard optimal confidence procedures are inadequate or even absurd, i.e., can provide confidence statements with a 95% empty or all-inclusive confidence set. A belt C(·) from a consistent nonexact-belt family, used with two confidence coefficients (γ = inf_θ P_θ [ θ ? C(X)] and γ⁺ = sup_θ P_θ[θ ? C(X)], is shown to provide a consistent nonexact-belt solution for estimating μ₂ -μ₁ in the Behrens-Fisher problem. A rule for consistent behaviour enables any confidence belt to be used consistently by providing each sample point with best upper and lower confidence levels [δ⁺(x) ≥ γ⁺, δ(x) ≤ γ], which give least-conservative consistent confidence statements ranging from practically exact through informative to noninformative. The rule also provides a consistency correction L(x) = δ⁺(x)-δ(X) enabling alternative confidence solutions to be compared on grounds of adequacy; this is demonstrated by comparing consistent conservative sample-point-wise solutions with inconsistent standard solutions for estimating μ₂/μ₁ (Creasy-Fieller-Neyman problem) and $\sqrt {\mu _1^2 + \mu _2^2 }$, a distance-estimation problem closely related to Stein's 1959 example     

On some optimal selection procedures for weibull populations

Consider k (k >(>)2) Weibull populations. We shall derive a method of constructing optimal selection procedures to select a subset of the k populations containing the best population which control the size of the selected subset and which maximises the minimum probability of making a correct selection. Procedures and results are derived for the case when sample sizes are unequal. Some tables and figures are given at the end of this paper.     

Determining the optimal allocation of testing resource for modular software system using dynamic programming

Abstract

Reliability is a major concern in the process of software development because unreliable software can cause failure in the computer system that can be hazardous. A way to enhance the reliability of software is to detect and remove the faults during the testing phase, which begins with module testing wherein modules are tested independently to remove a substantial number of faults within a limited resource. Therefore, the available resource must be allocated among the modules in such a way that the number of faults is removed as much as possible from each of the modules to achieve higher software reliability. In this article, we discuss the problem of optimal resource allocation of the testing resource for a modular software system, which maximizes the number of faults removed subject to the conditions that the amount of testing-effort is fixed, a certain percentage of faults is to be removed and a desired level of reliability is to be achieved. The problem is formulated as a non linear programming problem (NLPP), which is modeled by the inflection S-shaped software reliability growth models (SRGM) based on a non homogeneous Poisson process (NHPP) which incorporates the exponentiated Weibull (EW) testing-effort functions. A solution procedure is then developed using a dynamic programming technique to solve the NLPP. Furthermore, three special cases of optimum resource allocations are also discussed. Finally, numerical examples using three sets of software failure data are presented to illustrate the procedure developed and to validate the performance of the strategies proposed in this article. Experimental results indicate that the proposed strategies may be helpful to software project managers for making the best decisions in allocating the testing resource. In addition, the results are compared with those of Kapur et al. (2004), Huang and Lyu (2005), and Jha et al. (2010) that are available in the literature to deal the similar problems addressed in this article. It reveals that the proposed dynamic programming method for the testing-resource allocation problem yields a gain in efficiency over other methods.     

GROUPING PROBLEMS IN DISTRIBUTION-FREE REGRESSION1

In regression models having symmetric errors, exact distribution-free inference about individual parameters may be carried out by grouping observations, eliminating unwanted parameters within groups, and applying distribution free techniques for the symmetric location parameter problem. Models whose errors have identical but not symmetric distributions may obtain symmetry by taking differences between pairs of observations. Both grouping and differencing involve potential efficiency loss. The choice of an optimal scheme to minimize efficiency loss is expressible as a multi–assignment type of problem, whose solutions, exact and approximate, are discussed.     

An alternative proof of order statistics moment problem

A necessary and sufficient condition that two distributions having finite means are identical is that for any fixed integer r > 0, the expected values of their rth (n ? r) order statistics are equal [or the expected values of their (n-r)th (n > r ? 0) order statistics are equal] for all n where n is the sample size.     

Teaching Decision Theory Proof Strategies Using a Crowdsourcing Problem

Teaching how to derive minimax decision rules can be challenging because of the lack of examples that are simple enough to be used in the classroom. Motivated by this challenge, we provide a new example that illustrates the use of standard techniques in the derivation of optimal decision rules under the Bayes and minimax approaches. We discuss how to predict the value of an unknown quantity, θ ∈ {0, 1}, given the opinions of n experts. An important example of such crowdsourcing problem occurs in modern cosmology, where θ indicates whether a given galaxy is merging or not, and Y₁, …, Y_n are the opinions from n astronomers regarding θ. We use the obtained prediction rules to discuss advantages and disadvantages of the Bayes and minimax approaches to decision theory. The material presented here is intended to be taught to first-year graduate students.